In [1]}, a second measure of discrimination control is proposed, which replaces the denominator in Eq. (REF ) by \(p[Y = y]\)  and aims at making sure that the distribution of a target \(Y\)  given a sensitive attribute almost matches the target's. It is not hard to check from the proof of Lemma REF  that using the trivial fact \(p[Y = y] \le \max _i p[Y = y | A = a_i]\) , representation rate \( \tau \)  for \( P \in \mathcal {D}(\mathcal {X} \times \mathcal {Y} \times \mathcal {A}) \)  implies the same discrimination control as in Lemma REF . Such simple tricks would allow to show that the control of the representation rate brings guarantees on additional fairness measures, such as the maximal deviation between subgroups [2]}. Importantly, the bounds are not data-dependent, unlike [3]}.
